Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Witold Respondek Normandie Université, France INSA de Rouen, LMI ICMAT, Madrid, December 4, 2015
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Introduction 1 State-space equivalence and linearization 2 Feedback equivalence and linearization 3 Orbital feedback equivalence and linearization 4 Linearization via dynamic feedback and flatness 5 4 Definitions of flatness 6 Flat systems of minimal differential weight 7 Conclusions 8 2/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Introduction Summary Introduction 1 State-space equivalence and linearization 2 3 Feedback equivalence and linearization 4 Orbital feedback equivalence and linearization 5 Linearization via dynamic feedback and flatness 6 4 Definitions of flatness 7 Flat systems of minimal differential weight 8 Conclusions 3/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Introduction Class of control systems finite-dimensional smooth time-continous We will consider Ξ : ˙ x = F ( x , u ) x ∈ X , state space, an open subset of R n u ∈ U , set of control values, a subset of R m F is smooth ( C k or C ∞ ) with respect to ( x , u ) a control system is an underdetermined system of differential equations: n equations for n + m variables 4/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Introduction Very often: control-affine systems m x ∈ X ⊂ R n , u ∈ R m ∑ Σ : ˙ x = f ( x ) + u i g i ( x ) , i = 1 f and g 1 , . . . , g m are smooth vector fields on X state-dependent nonlinearities common in applications 5/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic Introduction Linearization problem When is Ξ or Σ equivalent (transformable) to a linear control system? define equivalence (or the class of transformations) find conditions for linearization construct linearizing transformations 6/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization Summary Introduction 1 State-space equivalence and linearization 2 3 Feedback equivalence and linearization 4 Orbital feedback equivalence and linearization 5 Linearization via dynamic feedback and flatness 6 4 Definitions of flatness 7 Flat systems of minimal differential weight 8 Conclusions 7/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization The system Ξ : ˙ x = F ( x , u ) , x ∈ X , u ∈ U and z = ˜ ˜ Ξ : ˙ F ( z , u ) , z ∈ Z , u ∈ U ( the same control ) are state-space equivalent, shortly S-equivalent, if there exists a diffeomorphism z = Φ ( x ) such that ∂ Φ ∂ x · F ( x , u ) = ˜ F ( Φ ( x ) , u ) i . e ., Φ ∗ F = ˜ F . the Jacobian matrix of Φ (the derivative of Φ , i.e, the tangent map of Φ ) maps the dynamics F of Ξ into ˜ F of ˜ Ξ A diffeomorphism is a map Φ such that Φ is bijective Φ and Φ − 1 are C k ( C ∞ ) A (local) diffeomorphism defines a (local) nonlinear change of coordinates z = Φ ( x ) 8/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization S-equivalence preserves trajectories X Z x(t,x 0 ,u) z(t,z 0 ,u) x 0 z 0 ϕ The image under Φ of a trajectory of Ξ is a trajectory of ˜ Ξ corresponding to the same control. 8/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization S-linearization Problem 1 When is Σ S-equivalent to a linear system, i.e., when does there exist z = Φ ( x ) transforming Σ into a linear system of the form m x ∈ R n ∑ z = Az + ˙ u i b i , i = 1 that is, for 1 ≤ i ≤ m , ∂ Φ ∂ Φ ∂ x ( x ) · f = Φ ∗ f = Az and ∂ x ( x ) · g i ( x ) = Φ ∗ g i = b i We want the same diffeomorphism Φ to transform f into Az (a linear vector field) and g i into b i , for 1 ≤ i ≤ m (constant vector fields) 9/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization Why is S-linearization interesting? If we want to solve a control problem for Σ and Σ is S-equivalent to a linear system Λ , then transform Σ into Λ solve the problem for the linear system Λ transform the solution (via the inverse Φ − 1 of Φ ) we identify intrinsic nonlinearities 10/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization A little bit of geometry: Lie bracket Given two vector fields f and g on X , we define their Lie bracket as [ f , g ]( x ) = ∂ g ∂ x ( x ) f ( x ) − ∂ f ∂ x ( x ) g ( x ) It is a new vector field on X . It is a geometric (invariant) object Φ ∗ [ f , g ] = [ Φ ∗ f , Φ ∗ g ] . It measures to what extent the flows of f and g do not commute 11/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization Define ad 0 = f g g ad f g = [ f , g ] [ f , ad k − 1 and, inductively, ad k = g ] = [ f , . . . , [ f , g ] , ] f g f For the single-input system x = f ( x ) + ug ( x ) ˙ the Lie bracket ad f g = [ f , g ] = [ f , f + g ] measures to what extent the trajectories of f (corresponding to u ≡ 0) do not commute with those of f + g (corresponding to u ≡ 1). 12/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization Theorem Σ is, locally around x 0 , S-equivalent to a controllable linear system Λ if and only if (SL1) span { ad q f g i ( x 0 ) : 1 ≤ i ≤ m , 0 ≤ q ≤ n − 1 } = R n (SL2) [ ad q f g i , ad r f g j ] = 0 , for 1 ≤ i , j ≤ m , 0 ≤ q , r ≤ n Interpretation (SL1) guarantees controllability of Λ (SL2) implies that all iterative Lie brackets containing at least two g i ’s vanish, i.e., [ L 0 , L 0 ] = 0, where L 0 is the strong accessibility Lie algebra. Verification (SL1) and (SL2) are verifiable in terms of f and g i ’s using differentiation and algebraic operations only (no need to solve PDE’s) 12/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization Theorem Σ on X is globally S-equivalent to a controllable linear system Λ on R n if and only if (SL1) span { ad q f g i ( x 0 ) : 1 ≤ i ≤ m , 0 ≤ q ≤ n − 1 } = R n (SL2) [ ad q f g i , ad r f g j ] = 0 , for 1 ≤ i , j ≤ m , 0 ≤ q , r ≤ n (SL3) the vector fields f, g 1 ,. . . ,g m are complete (equivalently, ad q f g i , 1 ≤ i ≤ m , 0 ≤ q ≤ n − 1 are complete). (SL4) X is simply connected If we drop (SL4), then Σ is globally S-equivalent to a controllable linear system Λ on T k × R n − k . 13/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization Constructing linearizing coordinates Assume, for simplicity, the scalar-input case m = 1. In order to find the linearizing diffeomorphism z = Φ ( x ) solve the system of n 1st order PDE’s: ( S ) ∂ Φ ∂ x A ( x ) = Id , where A ( x ) = ( A 1 ( x ) , . . . , A n ( x )) and A q ( x ) = ad q − 1 g ( x ) , for f 1 ≤ q ≤ n . (SL2) form the integrability conditions for (S) and assure the existence of solutions. 14/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization Do not confuse S-linearization with linear approximation Assume F ( x 0 , u 0 ) = 0. The linear approximation of ˙ x = F ( x , u ) is z ˙ = Az + Bv + higher order terms ˙ z = Az + Bv , where A = ∂ F ∂ x ( x 0 , u 0 ) and B = ∂ F ∂ u ( x 0 , u 0 ) So we neglect (erase) higher order terms In S-linearization higher order terms are compensated via the diffeomorphism Φ (no terms are neglected) 15/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization Consider the pendulum θ l m g The states are ( x 1 , x 2 ) = ( θ , ˙ θ ) and the control is the torque u 16/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization The equations are ˙ x 1 = x 2 − g 1 ˙ x 2 = l sin x 1 + ml 2 u . We have � 0 � 1 � � � � x 2 ml 2 f = , g = , ad f g = − − g 1 l sin x 1 0 ml 2 yielding [ g , ad f g ] = 0 but [ ad f g , ad 2 f g ] � = 0 which implies that the pendulum is not S-linearizable 17/ 75
Linearization of nonlinear control systems: state-space, feedback, orbital, and dynamic State-space equivalence and linearization But put u = ml 2 ( g l sin x 1 + v ) we get the linear controllable system (in the Brunovsky form) x 1 ˙ = x 2 ˙ = x 2 v . therefore there are systems that become linear after applying a (nonlinear) transformation in the control space 18/ 75
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